On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

Sierpi\'nski graphs $S_p^n$ form an extensively studied family of graphsof fractal nature applicable in topology, mathematics of the Tower of Hanoi,computer science, and elsewhere. An almost-extreme vertex of $S_p^n$ isintroduced as a vertex that is either adjacent to an extreme vertex of$S_p^n$ or is incident to an edge between two subgraphs of $S_p^{n}$isomorphic to $S_p^{n-1}$. Explicit formulas are given for thedistance in $S_p^{n}$ between an arbitrary vertex and an almost-extremevertex. The formulas are applied to compute the total distance ofalmost-extreme vertices and to obtain the metric dimension of Sierpi\'nski graphs